scholarly journals The Group Action on the Finite Projective Planes of Orders 29, 31, 32, 37

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm Abdulzahra Makhrib Al-Seraji ◽  
Mohammed Abdul Hadi Sarhan
Keyword(s):  
Projective Plane ◽  
Group Action ◽  
Three Dimensional ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Singer Group ◽  
Group Effects

The goal of this research is to study the group effects on a projective plane PG (2,q), when is not a prime, and then describe the geometry of these orbits by Singer group for values of q=29,31,32,37 . Also, to establish three dimensional codes and arcs and study the properties of subsets in a projective plane of order q.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

Finite Projective Planes with Affine Subplanes

1964 ◽  
Vol 7 (4) ◽  
pp. 549-559 ◽  
Author(s):  
T. G. Ostrom ◽  
F. A. Sherk
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Desarguesian Projective Planes

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

scholarly journals On Unitary Polarities of Finite Projective Planes

1971 ◽  
Vol 23 (6) ◽  
pp. 1060-1077 ◽  
Author(s):  
William M. Kantor
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Absolute Point ◽  
Finite Projective Planes ◽  
Unitary Polarity

A unitary polarity of a finite projective plane of order q2 is a polarity θ having q3 + 1 absolute points and such that each nonabsolute line contains precisely q + 1 absolute points. Let G(θ) be the group of collineations of centralizing θ. In [15] and [16], A. Hoffer considered restrictions on G(θ) which force to be desarguesian. The present paper is a continuation of Hoffer's work. The following are our main results.THEOREM I. Let θ be a unitary polarity of a finite projective planeof order q2. Suppose that Γ is a subgroup of G(θ) transitive on the pairs x, X, with x an absolute point and X a nonabsolute line containing x. Thenis desarguesian and Γ contains PSU(3, q).

On Baer Involutions of Finite Projective Planes

1970 ◽  
Vol 22 (4) ◽  
pp. 878-880 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Finite Projective Planes ◽  
Special Cases ◽  
Collineation Groups ◽  
Baer Involution ◽  
Finite Desarguesian Projective Plane

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

scholarly journals On some hyperbolic planes from finite projective planes

2001 ◽  
Vol 25 (12) ◽  
pp. 757-762 ◽  
Author(s):  
Basri Celik
Keyword(s):  
Projective Plane ◽  
Hyperbolic Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Baer Subplane ◽  
Combinatorial Properties ◽  
Finite Projective Planes ◽  
Hyperbolic Planes

LetΠ=(P,L,I)be a finite projective plane of ordern, and letΠ′=(P′,L′,I′)be a subplane ofΠwith ordermwhich is not a Baer subplane (i.e.,n≥m2+m). Consider the substructureΠ0=(P0,L0,I0)withP0=P\{X∈P|XIl,  l∈L′},L0=L\L′whereI0stands for the restriction ofItoP0×L0. It is shown that everyΠ0is a hyperbolic plane, in the sense of Graves, ifn≥m2+m+1+m2+m+2. Also we give some combinatorial properties of the line classes inΠ0hyperbolic planes, and some relations between its points and lines.

The Non-Existence of Finite Projective Planes of Order 10

1989 ◽  
Vol 41 (6) ◽  
pp. 1117-1123 ◽  
Author(s):  
C. W. H. Lam ◽  
L. Thiel ◽  
S. Swiercz
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Finite Projective Planes

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that1. every line contains n + 1 points,2. every point is on n + 1 lines,3. any two distinct lines intersect at exactly one point, and4. any two distinct points lie on exactly one line.It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

Affine Subplanes of Finite Projective Planes

1965 ◽  
Vol 17 ◽  
pp. 977-1009 ◽  
Author(s):  
J. F. Rigby
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Finite Projective Planes

Let π be a finite projective plane of order n containing a finite projective subplane π* of order u < n. Bruck has shown (1, p. 398) that if π contains a point that does not lie on any line of π*, then n ≥ u2 + u, while if every point of π lies on a line of π* then n = u2.Let π be a finite projective plane of order n containing a finite affine subplane π0 of order m < n.

The group action on the finite projective planes of orders 53, 61, 64

2020 ◽  
Vol 23 (8) ◽  
pp. 1573-1582
Author(s):  
Najm Abdulzahra Makhrib Al-Seraji ◽  
Esam A. Alnussairy ◽  
Zainab Sadiq Jafar
Keyword(s):  
Group Action ◽  
Projective Planes ◽  
Finite Projective Planes

scholarly journals Embedding Cycles in Finite Planes

10.37236/3377 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Felix Lazebnik ◽  
Keith E. Mellinger ◽  
Oscar Vega
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Finite Affine ◽  
Finite Planes

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$,  a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$,  a $k$-cycle can be embedded in any projective plane of order $q$.

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